Hello,

If you failed your first compartment attempt, you typically have the option to reappear for the compartment exam in the next session. However, whether you can simultaneously appear for both compartment and improvement exams as a private candidate depends on the regulations of the specific education board (such as CBSE, ICSE, or state boards).

For CBSE, generally:

1. Compartment Exam: If you fail in the first compartment attempt, you can appear for the second compartment exam.

2. Improvement Exam: This exam is usually for students who have passed but wish to improve their marks in specific subjects.

According to CBSE rules, you usually cannot take the improvement exam for a subject in which you have not yet passed. Therefore, if you still need to clear a compartment subject, you must first pass it before you can apply for an improvement exam in that subject.

To confirm and get the most accurate information, it's best to check the official guidelines from your education board or contact them directly.

Hope it helps !